Abstract

A generalization of the well-known Fibonacci sequence is the k−Fibonacci sequence whose first k terms are 0,…,0,1 and each term afterwards is the sum of the preceding k terms. In this paper, we find all k-Fibonacci numbers that are curious numbers (i.e., numbers whose base ten representation have the form a⋯ab⋯ba⋯a). This work continues and extends the prior result of Trojovský, who found all Fibonacci numbers with a prescribed block of digits, and the result of Alahmadi et al., who searched for k-Fibonacci numbers, which are concatenation of two repdigits.

Highlights

  • Given a couple of non-negative integers and m, we shall define the (, m)−curious number as a natural number with the following base ten representationCitation: Herrera, J.L.; Bravo, J.J.; Gómez, C.A

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  • We present some basic properties of the k−Fibonacci sequence and give some important estimations needed for the sequel

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Summary

Introduction

Given a couple of non-negative integers and m, we shall define the ( , m)−curious number as a natural number with the following base ten representation. It should be mentioned that Luca [3] in 2000 showed that 55 and 11 are the largest repdigits in the Fibonacci and Lucas sequences, respectively Since this result has been generalized and extended in various directions. Alahmadi et al [10] generalized recently the results mentioned above by showing that only repdigits with at least two digits as a product of consecutive k-Fibonacci numbers occur only for (k, ) = (2, 1), (3, 1), extending the works [11,12], which dealt with the particular cases of Fibonacci and Tribonacci numbers. Before presenting our main theorem, it is important to mention that in Equation (1), we assumed , m ≥ 1 and a = b since otherwise, the problem reduces to finding all k−Fibonacci numbers that are repdigits or concatenations of two repdigits; these problems have been already solved in [9,18] (see [20]).

Preliminary Results
On k–Fibonacci Numbers
Linear Forms in Logarithms
Reduction Tools
Proof Theorem 1
Powers of 2 Which Are Curious Numbers
Bounding n in Terms of k
An Inequality for in Terms of k
The Case of Small k
9: ]; References
Full Text
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